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We study a real-time bidding problem resulting from a set of contractualobligations stipulating that a firm win a specified number of heterogeneousimpressions or ad placements over a defined duration in a real-time auction.The contracts specify item targeting criteria (which may be overlapping), and asupply requirement. Using the Pontryagin maximum principle, we show that theresulting continuous time and time inhomogeneous planning problem can bereduced into a finite dimensional convex optimization problem and solved tooptimality. In addition, we provide algorithms to update the bidding plan overtime via a receding horizon. Finally, we provide numerical results based onreal data and show a connection to production-transportation problems.

Optimal Real-time Bidding Policies for Contract Fulfillment in Second Price Auctions